problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
What is the 7th oblong number? | 56 | The 7th oblong number is the number of dots in a rectangular grid of dots with 7 columns and 8 rows. Thus, the 7th oblong number is $7 \times 8=56$. | 1 | 362.4375 | 362.4375 | -1 |
There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square? | \frac{2}{5} | 1. **Identify the total number of points and the position of $P$:**
The square contains $81$ grid points, arranged in a $9 \times 9$ grid. Point $P$, being at the center, is located at the center grid point.
2. **Determine the number of points excluding $P$:**
Excluding point $P$, there are $81 - 1 = 80$ points.... | 0.5 | 6,078.3125 | 4,702.375 | 7,454.25 |
Find the simplest method to solve the system of equations using substitution $$\begin{cases} x=2y\textcircled{1} \\ 2x-y=5\textcircled{2} \end{cases}$$ | y = \frac{5}{3} | 0.3125 | 3,960.3125 | 3,924.6 | 3,976.545455 | |
Given that $a$ is a multiple of $1428$, find the greatest common divisor of $a^2+9a+24$ and $a+4$. | 4 | 0.875 | 4,715.3125 | 4,218.642857 | 8,192 | |
In a right triangular pyramid P-ABC, where PA, PB, and PC are mutually perpendicular and PA=1, the center of the circumscribed sphere is O. Find the distance from O to plane ABC. | \frac{\sqrt{3}}{6} | 0 | 7,657.8125 | -1 | 7,657.8125 | |
What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\text{th}}$ roots of unity? | 12 | 0.75 | 6,122.9375 | 5,433.25 | 8,192 | |
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born? | 1979 | 1. **Identify the year of the seventh AMC 8**:
The first AMC 8 was given in 1985. Since the AMC 8 is held annually, the subsequent AMC 8s would be in the years 1986, 1987, 1988, 1989, 1990, and the seventh AMC 8 would be in 1991.
2. **Determine Samantha's age during the seventh AMC 8**:
It is given that Samanth... | 0.9375 | 2,609.0625 | 2,236.866667 | 8,192 |
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region? | 54\sqrt{3}-18\pi | 1. **Calculate the area of the hexagon**:
A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is given as 6. The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
Substituting \( s = 6 \):
\[
... | 0.0625 | 4,718.0625 | 5,400 | 4,672.6 |
In $\triangle ABC$, the sides corresponding to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a=2$, $b=3$, $\cos B= \frac{1}{3}$.
(1) Find the value of side $c$;
(2) Find the value of $\cos (A-C)$. | \frac{23}{27} | 0.875 | 4,535.25 | 4,391.857143 | 5,539 | |
A regular $2018$ -gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$ -gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ( $2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals ... | 2 \times 1008! | 0 | 8,192 | -1 | 8,192 | |
Given \(\alpha, \beta \in \left[0, \frac{\pi}{4}\right]\), find the maximum value of \(\sin(\alpha - \beta) + 2 \sin(\alpha + \beta)\). | \sqrt{5} | 0.375 | 7,545.9375 | 6,469.166667 | 8,192 | |
Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$. | 12^{1/7} | 0 | 2,489 | -1 | 2,489 | |
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \). | 1987 | 0.0625 | 7,891.8125 | 3,389 | 8,192 | |
Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$ , $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (f... | 11 | 0.0625 | 8,167.875 | 7,806 | 8,192 | |
How many positive divisors of $150$ are not divisible by 5? | 4 | 1 | 3,623.0625 | 3,623.0625 | -1 | |
Given \(\theta = \arctan \frac{5}{12}\), find the principal value of the argument of the complex number \(z = \frac{\cos 2\theta + i \sin 2\theta}{239 + i}\). | \frac{\pi}{4} | 0.5 | 6,851.4375 | 5,510.875 | 8,192 | |
The coach of the math training team needs to photocopy a set of materials for 23 team members. The on-campus copy shop charges 1.5 yuan per page for the first 300 pages and 1 yuan per page for any additional pages. The cost of photocopying these 23 sets of materials together is exactly 20 times the cost of photocopying... | 950 | 0.125 | 7,996.1875 | 6,625.5 | 8,192 | |
What is the perimeter of the triangle formed by the points of tangency of the incircle of a 5-7-8 triangle with its sides? | \frac{9 \sqrt{21}}{7}+3 | Let $\triangle A B C$ be a triangle with sides $a=7, b=5$, and $c=8$. Let the incircle of $\triangle A B C$ be tangent to sides $B C, C A$, and $A B$ at points $D, E$, and $F$. By the law of cosines (using the form $\left.\cos (A)=\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)$, we have $$\begin{aligned} & \cos (A)=\frac{8^{2}... | 0 | 8,192 | -1 | 8,192 |
In an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), the angles \(ABD\) and \(DBC\) are \(135^{\circ}\) and \(15^{\circ}\) respectively, and \(BD = \sqrt{6}\). Find the perimeter of the trapezoid. | 9 - \sqrt{3} | 0.0625 | 7,994.9375 | 8,192 | 7,981.8 | |
In a box, there are 10 balls of the same size, among which 3 are labeled with 1, 4 are labeled with 2, and 3 are labeled with 5. First, a ball is randomly drawn from the box and then put back. After that, another ball is randomly drawn (assuming the probability of drawing each ball is the same). Let the sum of the labe... | 5.2 | 0.1875 | 4,316.6875 | 4,061.666667 | 4,375.538462 | |
Positive integers $a$, $b$, $c$, and $d$ are such that $a<b<c<d$, and the system of equations
\[ 2x + y = 2007 \quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d| \]
has exactly one solution. What is the minimum value of $d$? | 504 | 0 | 8,192 | -1 | 8,192 | |
Given the ellipse $C\_1: \frac{x^2}{8} + \frac{y^2}{4} = 1$ with left and right foci $F\_1$ and $F\_2$, a line $l\_1$ is drawn through point $F\_1$ perpendicular to the $x$-axis. Line $l\_2$ is perpendicular to $l\_1$ at point $P$, and the perpendicular bisector of segment $PF\_2$ intersects $l\_2$ at point $M$.
(I) F... | \frac{64}{9} | 0 | 8,192 | -1 | 8,192 | |
How many numbers are in the list starting from $-48$, increasing by $7$ each time, up to and including $119$? | 24 | 0.8125 | 2,121.5625 | 2,432.307692 | 775 | |
Given the function $f(x)=\begin{cases} (\frac{1}{2})^{x} & x\geqslant 4 \\ f(x+1) & x < 4 \end{cases}$, find the value of $f(2+\log_{2}3)$. | \frac{1}{24} | 0.6875 | 6,109.25 | 5,844 | 6,692.8 | |
A circular cylindrical post with a circumference of 4 feet has a string wrapped around it, spiraling from the bottom of the post to the top of the post. The string evenly loops around the post exactly four full times, starting at the bottom edge and finishing at the top edge. The height of the post is 12 feet. What is ... | 20 | 0.9375 | 2,042.75 | 1,632.8 | 8,192 | |
Given the diameter $d=\sqrt[3]{\dfrac{16}{9}V}$, find the volume $V$ of the sphere with a radius of $\dfrac{1}{3}$. | \frac{1}{6} | 0.8125 | 5,886.9375 | 5,355 | 8,192 | |
Given \( x \in \mathbb{R} \), find the maximum value of \(\frac{\sin x(2-\cos x)}{5-4 \cos x}\). | \frac{\sqrt{3}}{4} | 0 | 7,767.375 | -1 | 7,767.375 | |
Given a permutation $\pi$ of the set $\{1,2, \ldots, 10\}$, define a rotated cycle as a set of three integers $i, j, k$ such that $i<j<k$ and $\pi(j)<\pi(k)<\pi(i)$. What is the total number of rotated cycles over all permutations $\pi$ of the set $\{1,2, \ldots, 10\}$ ? | 72576000 | Let us consider a triple $(i, j, k)$ with $i<j<k$ and determine how many permutations rotate it. There are $\binom{10}{3}$ choices for the values of $\pi(i), \pi(j), \pi(k)$ and the choice of this set of three determines the values of $\pi(i), \pi(j), \pi(k)$. The other 7 values then have 7 ! ways to be arranged (any p... | 0.125 | 7,887.9375 | 5,759.5 | 8,192 |
If four people, A, B, C, and D, line up in a row, calculate the number of arrangements in which B and C are on the same side of A. | 16 | 0.375 | 7,730.3125 | 6,960.833333 | 8,192 | |
Another professor enters the same room and says, 'Each of you has to write down an integer between 0 and 200. I will then compute $X$, the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number closest to $X$ (either above or below $X$)... | 7 | Use the same logic to get 7. Note 6 and 8 do not work. | 0.375 | 7,034.625 | 5,105.666667 | 8,192 |
Determine the number of six-digit palindromes. | 9000 | 0 | 2,339.3125 | -1 | 2,339.3125 | |
After learning about functions, the mathematics team of a high school first grade conducted a mathematical modeling activity. Through a survey of the sales of a certain product in a supermarket near the school, it was found that the relationship between the daily sales price P(x) (in yuan per item) of the product in th... | 121 | 0.75 | 5,837 | 5,519.5 | 6,789.5 | |
Find $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$12\sqrt{2}$",R/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy] | 12\sqrt{2} | 0.125 | 4,268.625 | 3,524.5 | 4,374.928571 | |
If for a number \( x \) you calculate the sum of its digits and repeat this process two more times with the resulting number, you get a sequence of four numbers. Find the smallest \( x \) for which all four numbers are distinct and the last number is 2. | 2999 | 0.1875 | 7,993.4375 | 7,133 | 8,192 | |
Three vertices of a rectangle are at points $(2, 7)$, $(13, 7)$, and $(13, -6)$. What is the area of the intersection between this rectangle and the circular region described by equation $(x - 2)^2 + (y + 6)^2 = 25$? | \frac{25}{4}\pi | 0.4375 | 6,739.125 | 6,242.142857 | 7,125.666667 | |
Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 20$ and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = xyz.\]Find $\frac{x^3 + y^3 + z^3}{xyz}.$ | 13 | 1 | 3,212.5625 | 3,212.5625 | -1 | |
In triangle $ABC,$ $\angle B = 30^\circ,$ $AB = 150,$ and $AC = 50 \sqrt{3}.$ Find the sum of all possible values of $BC.$ | 150 \sqrt{3} | 0.875 | 4,536.3125 | 4,014.071429 | 8,192 | |
A bouncy ball is dropped from a height of 100 meters. After each bounce, it reaches a height that is half of the previous one. What is the total distance the ball has traveled when it hits the ground for the 10th time? (Round the answer to the nearest whole number) | 300 | 0.4375 | 7,008.9375 | 5,487.857143 | 8,192 | |
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ are perpendicular. If $AD= 15$ and $BE = 20$, then what is the area of $\triangle ABC$? | 200 | 0.875 | 5,390.6875 | 4,990.5 | 8,192 | |
Let $a, a', b,$ and $b'$ be real numbers with $a$ and $a'$ nonzero. The solution to $ax+b=0$ is less than the solution to $a'x+b'=0$ if and only if | $\frac{b'}{a'}<\frac{b}{a}$ | 1. **Identify the solutions to the equations**:
The solution to the equation $ax + b = 0$ can be found by isolating $x$:
\[
ax = -b \implies x = \frac{-b}{a}.
\]
Similarly, the solution to the equation $a'x + b' = 0$ is:
\[
a'x = -b' \implies x = \frac{-b'}{a'}.
\]
2. **Set up the inequality c... | 0 | 7,917.75 | -1 | 7,917.75 |
In $\triangle ABC$, points $D$ and $E$ lie on $AB$, as shown. If $AD=DE=EB=CD=CE$, what is the measure of $\angle ABC$? | 30^{\circ} | Since $CD=DE=EC$, then $\triangle CDE$ is equilateral, which means that $\angle DEC=60^{\circ}$. Since $\angle DEB$ is a straight angle, then $\angle CEB=180^{\circ}-\angle DEC=180^{\circ}-60^{\circ}=120^{\circ}$. Since $CE=EB$, then $\triangle CEB$ is isosceles with $\angle ECB=\angle EBC$. Since $\angle ECB+\angle CE... | 0.8125 | 5,590.375 | 4,992.461538 | 8,181.333333 |
Let $g(x)$ be the function defined on $-2 \leq x \leq 2$ by the formula $$g(x) = 2 - \sqrt{4-x^2}.$$ This is a vertically stretched version of the previously given function. If a graph of $x=g(y)$ is overlaid on the graph of $y=g(x)$, then one fully enclosed region is formed by the two graphs. What is the area of that ... | 2.28 | 0.375 | 7,742.0625 | 7,389.5 | 7,953.6 | |
Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ a... | 389 | Consider a 13-14-15 triangle. $A=84.$ [By Heron's Formula or by 5-12-13 and 9-12-15 right triangles.]
The inradius is $r=\frac{A}{s}=\frac{84}{21}=4$, where $s$ is the semiperimeter. Scale the triangle with the inradius by a linear scale factor, $u.$
The circumradius is $R=\frac{abc}{4rs}=\frac{13\cdot 14\cdot 15}{4\... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\infty)$, determine the value of $m$. | -1 | 0 | 8,192 | -1 | 8,192 | |
As a result of measuring the four sides and one of the diagonals of a certain quadrilateral, the following numbers were obtained: $1 ; 2 ; 2.8 ; 5 ; 7.5$. What is the length of the measured diagonal? | 2.8 | 0.625 | 5,763.25 | 4,306 | 8,192 | |
Let the function \( f(x) = x^2 - x + 1 \). Define \( f^{(n)}(x) \) as follows:
$$
f^{(1)}(x) = f(x), \quad f^{(n)}(x) = f\left(f^{(n-1)}(x)\right).
$$
Let \( r_{n} \) be the arithmetic mean of all the roots of \( f^{(n)}(x) = 0 \). Find \( r_{2015} \). | \frac{1}{2} | 0.375 | 7,482.4375 | 6,299.833333 | 8,192 | |
How many 7-digit numbers divisible by 9 are there, where the second to last digit is 5? | 100000 | 0.4375 | 7,255.875 | 6,916.142857 | 7,520.111111 | |
In $\triangle ABC$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$.
(1) Find the measure of angle $A$;
(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, if $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{6} | 0 | 6,810.125 | -1 | 6,810.125 | |
Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip? | \frac{300}{7} | 1. Let the total distance of Randy's trip be denoted as $x$ miles.
2. According to the problem, Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. This can be expressed as:
\[
\frac{x}{3} + 20 + \frac{x}{5} = x
\]
3. To simplify ... | 0.8125 | 4,621.25 | 3,797.230769 | 8,192 |
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. | 20 | 1. **Calculate the time taken by each swimmer to swim the length of the pool:**
- The first swimmer swims at $3$ feet per second, so the time to swim $90$ feet is:
\[
\frac{90 \text{ feet}}{3 \text{ feet/second}} = 30 \text{ seconds}
\]
- The second swimmer swims at $2$ feet per second, so the time... | 0 | 8,139 | -1 | 8,139 |
Marcus has two numbers, $a$ and $b$. When he divides $a$ by 45 he gets a remainder of 37. When he divides $b$ by 30 he gets a remainder of $9$. What remainder does he get when he divides $a+b$ by 15? | 1 | 1 | 1,838.1875 | 1,838.1875 | -1 | |
Calculate:<br/>$(1)-4\times 9$;<br/>$(2)10-14-\left(-5\right)$;<br/>$(3)-3×(-\frac{1}{3})^3$;<br/>$(4)-56+(-8)×(\frac{1}{8})$. | -57 | 1 | 589.25 | 589.25 | -1 | |
Given that $α \in (0, \frac{π}{2})$ and $\sin (\frac{π}{4} - α) = \frac{\sqrt{10}}{10}$,
(1) find the value of $\tan 2α$;
(2) find the value of $\frac{\sin (α + \frac{π}{4})}{\sin 2α + \cos 2α + 1}$. | \frac{\sqrt{10}}{8} | 0 | 4,128.8125 | -1 | 4,128.8125 | |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$, with $A$ being the left vertex, and $P$ is a point on the ellipse $E$ such that the circle with diameter $PF_1$ passes through $F_2$ and $|PF_{2}|= \frac {1}{4}|AF_{2}|$, determine the eccentricity of the ellipse $E$. | \frac{3}{4} | 0.6875 | 6,379.3125 | 5,555.363636 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m}=(\cos A,\cos B)$ and $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) Find the maximum val... | \sqrt {3} | 0 | 5,259.375 | -1 | 5,259.375 | |
Marie has 75 raspberry lollipops, 132 mint lollipops, 9 blueberry lollipops, and 315 coconut lollipops. She decides to distribute these lollipops equally among her 13 friends, distributing as many as possible. How many lollipops does Marie end up keeping for herself? | 11 | 0.8125 | 2,820.875 | 2,697.461538 | 3,355.666667 | |
If $f\left(x\right)=\ln |a+\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____. | \ln 2 | 0.9375 | 4,155.0625 | 3,885.933333 | 8,192 | |
A factory has two branches, one in location A and the other in location B, producing 12 and 6 machines respectively. Now, they need to distribute 10 machines to area A and 8 machines to area B. It is known that the transportation cost for moving one machine from location A to area A and B is 400 and 800 yuan respective... | 8600 | 0.375 | 6,183 | 5,466.666667 | 6,612.8 | |
What is the greatest number of Sundays that can occur in the first $49$ days of a year? | 7 | 0.5625 | 6,930.1875 | 5,948.777778 | 8,192 | |
The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]
fill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray);
fill( Circle((1,1), 1.2), white);
fill( C... | 36 - 12\sqrt{3} - 4\pi | 0 | 6,304.3125 | -1 | 6,304.3125 | |
In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ). | 19 | 0 | 8,192 | -1 | 8,192 | |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the condition: for any real numbers \( x \) and \( y \),
\[ f(2x) + f(2y) = f(x+y) f(x-y) \]
and given that \( f(\pi) = 0 \) and \( f(x) \) is not identically zero, determine the period of \( f(x) \). | 4\pi | 0.125 | 7,435.625 | 3,593 | 7,984.571429 | |
There are 12 different-colored crayons in a box. How many ways can Karl select four crayons if the order in which he draws them out does not matter? | 495 | 1 | 1,279.125 | 1,279.125 | -1 | |
Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given). | \frac{2015}{2} |
To find the roots of the equation \((x-a)(x-b)=(x-c)(x-d)\), given that \(a + d = b + c = 2015\) and \(a \neq c\), we will simplify the equation and determine the solutions.
### Step 1: Expand Both Sides
Expanding both sides of the equation, we have:
\[
(x-a)(x-b) = x^2 - (a+b)x + ab
\]
\[
(x-c)(x-d) = x^2 - (c+d)x +... | 0.6875 | 5,853.625 | 4,790.727273 | 8,192 |
In her last basketball game, Jackie scored 36 points. These points raised the average number of points that she scored per game from 20 to 21. To raise this average to 22 points, how many points must Jackie score in her next game? | 38 | Suppose that Jackie had played $n$ games before her last game. Since she scored an average of 20 points per game over these $n$ games, then she scored $20n$ points over these $n$ games. In her last game, she scored 36 points and so she has now scored $20n+36$ points in total. But, after her last game, she has now playe... | 1 | 2,768.125 | 2,768.125 | -1 |
Three workshops A, B, and C of a factory produced the same kind of product, with quantities of 120, 60, and 30, respectively. To determine if there are significant differences in product quality, a stratified sampling method was used to take a sample of size n for inspection, with 2 samples taken from workshop B.
(I) ... | \frac{2}{3} | 1 | 3,734.625 | 3,734.625 | -1 | |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=2\cos\varphi}\\{y=\sqrt{2}\sin\varphi}\end{array}}\right.$ (where $\varphi$ is the parameter). Taking point $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordina... | \frac{2\sqrt{7}}{3} | 0 | 6,695.9375 | -1 | 6,695.9375 | |
In the quadrilateral pyramid \( P-ABCD \), it is given that \( AB \parallel CD \), \( AB \perp AD \), \( AB = 4 \), \( AD = 2\sqrt{2} \), \( CD = 2 \), and \( PA \perp \) plane \( ABCD \) with \( PA = 4 \). Let \( Q \) be a point on the segment \( PB \), and the sine of the angle between the line \( QC \) and the plane... | 1/2 | 0 | 6,687.8125 | -1 | 6,687.8125 | |
How many distinct, natural-number factors does $4^3 \cdot 5^4 \cdot 6^2$ have? | 135 | 1 | 2,037.375 | 2,037.375 | -1 | |
Subset \( S \subseteq \{1, 2, 3, \ldots, 1000\} \) is such that if \( m \) and \( n \) are distinct elements of \( S \), then \( m + n \) does not belong to \( S \). What is the largest possible number of elements in \( S \)? | 501 | 0 | 7,948.25 | -1 | 7,948.25 | |
There are 7 volunteers, among which 3 people only speak Russian, and 4 people speak both Russian and English. From these, 4 people are to be selected to serve as translators for the opening ceremony of the "Belt and Road" summit, with 2 people serving as English translators and 2 people serving as Russian translators. ... | 60 | 0.25 | 7,070.375 | 6,537.25 | 7,248.083333 | |
If the graph of the function $f(x)=\sin \omega x+\sin (\omega x- \frac {\pi}{2})$ ($\omega > 0$) is symmetric about the point $\left( \frac {\pi}{8},0\right)$, and there is a zero point within $\left(- \frac {\pi}{4},0\right)$, determine the minimum value of $\omega$. | 10 | 0.0625 | 7,994.8125 | 8,192 | 7,981.666667 | |
Find the eighth term of the sequence $1440,$ $1716,$ $1848,\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences. | 348 | If you multiply the corresponding terms of two arithmetic sequences, you get the terms of a quadratic function. Thus, we have a quadratic $ax^2+bx+c$ such that $f(1)=1440$, $f(2)=1716$, and $f(3)=1848$. Plugging in the values for x gives us a system of three equations:
$a+b+c=1440$
$4a+2b+c=1716$
$9a+3b+c=1848$
Solvin... | 0.625 | 7,381.375 | 6,895 | 8,192 |
Matrices $A$ , $B$ are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0 1 & 2 & 0 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 2 & 4 & 0 0 & 0 & 12\end{pmatrix}\]
Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ . | \frac{\pi}{3} | 0.375 | 6,764.25 | 4,384.666667 | 8,192 | |
(1) Given a sequence $\{a_n\}$ that satisfies $a_1a_2…a_n=n+1$, find $a_3=$
(2) Let $\overrightarrow{e_1}, \overrightarrow{e_2}$ be unit vectors, where $\overrightarrow{a}=2\overrightarrow{e_1}+\overrightarrow{e_2}, \overrightarrow{b}=\overrightarrow{e_2}$, and $\overrightarrow{a} \cdot \overrightarrow{b}=2$, find $|\... | 6:5:4 | 0.5 | 7,606.6875 | 7,021.375 | 8,192 | |
Given the variables $a$ and $b$ satisfying $b=-\frac{1}{2}a^2+3\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$. | \frac{9}{5} | 0.375 | 7,342.1875 | 6,213.5 | 8,019.4 | |
Let the function $f(x)= \begin{cases} \log_{2}(1-x) & (x < 0) \\ g(x)+1 & (x > 0) \end{cases}$. If $f(x)$ is an odd function, determine the value of $g(3)$. | -3 | 1 | 1,663.125 | 1,663.125 | -1 | |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the... | \frac{2\sqrt{3} + 1}{3} | 0 | 6,783.625 | -1 | 6,783.625 | |
Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and that $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^{2}+y^{2}$. | -90 | We have $x^{2}+y^{2}+2 x y=1$. Define $a=2 x y$ and $b=x^{2}+y^{2}$ for convenience. Then $a+b=1$ and $b-a=x^{2}+y^{2}-2 x y=(x-y)^{2}=2 b-1$ so that $x, y=\frac{\sqrt{2 b-1} \pm 1}{2}$. Then $x^{20}+y^{20}=\left(\frac{\sqrt{2 b-1}+1}{2}\right)^{20}+\left(\frac{\sqrt{2 b-1}-1}{2}\right)^{20}=\frac{1}{2^{20}}\left[(\sqr... | 0 | 7,840.625 | -1 | 7,840.625 |
Given that point $C$ is the midpoint between points $A$ and $B$. At 7:00 AM, Car 1 departs from $A$ towards $B$, Car 2 departs from $B$ towards $A$, and Car 3 departs from $C$ towards $A$. When Car 1 and Car 3 meet, Car 2 has traveled exactly $\frac{3}{8}$ of the total distance. At 10:00 AM, Car 3 reaches point $A$, an... | 336 | 0 | 8,192 | -1 | 8,192 | |
What is the value of $k$ if the side lengths of four squares are shown, and the area of the fifth square is $k$? | 36 | Let $s$ be the side length of the square with area $k$.
The sum of the heights of the squares on the right side is $3+8=11$.
The sum of the heights of the squares on the left side is $1+s+4=s+5$.
Since the two sums are equal, then $s+5=11$, and so $s=6$.
Therefore, the square with area $k$ has side length 6, an... | 0 | 7,091.75 | -1 | 7,091.75 |
A marathon is $26$ miles and $385$ yards. One mile equals $1760$ yards.
Leila has run ten marathons in her life. If the total distance Leila covered in these marathons is $m$ miles and $y$ yards, where $0\le y<1760$, what is the value of $y$? | 330 | 0.75 | 4,540.5625 | 3,323.416667 | 8,192 | |
Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $1$, let $P$ be a moving point on the space diagonal $B C_{1}$ and $Q$ be a moving point on the base $A B C D$. Find the minimum value of $D_{1} P + P Q$. | 1 + \frac{\sqrt{2}}{2} | 0 | 7,997.75 | -1 | 7,997.75 | |
Find the best approximation of $\sqrt{3}$ by a rational number with denominator less than or equal to $15$ | \frac{26}{15} | 0 | 8,192 | -1 | 8,192 | |
A convex polyhedron \(Q\) has \(30\) vertices, \(70\) edges, and \(40\) faces, of which \(30\) are triangular and \(10\) are pentagonal. Compute the number of space diagonals in polyhedron \(Q\). | 315 | 0.75 | 7,067.0625 | 6,692.083333 | 8,192 | |
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(2-x) = f(x)$, and it is monotonically decreasing on the interval $[0, 1)$. If the equation $f(x) = -1$ has real roots in the interval $[0, 1)$, then the sum of all real roots of the equation $f(x) = 1$ in the interval $[-1, 7]$ is $\_\_\_\_\_\_$. | 12 | 0 | 8,192 | -1 | 8,192 | |
Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$ . Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ | $\pi/2$ | 0 | 7,984.8125 | -1 | 7,984.8125 | |
Given four real numbers that form an arithmetic sequence: -9, $a_1$, $a_2$, -1, and five real numbers that form a geometric sequence: -9, $b_1$, $b_2$, $b_3$, -1, find the value of $b_2(a_2-a_1)$. | -8 | 0.875 | 3,702.875 | 3,489.5 | 5,196.5 | |
The radius of the circumcircle of an acute triangle \( ABC \) is 1. It is known that the center of the circle passing through the vertices \( A \), \( C \), and the orthocenter of triangle \( ABC \) lies on this circumcircle. Find the length of side \( AC \). | \sqrt{3} | 0.0625 | 8,159.1875 | 7,667 | 8,192 | |
Given the function $f\left( x \right)=2\sin (\omega x+\varphi )\left( \omega \gt 0,\left| \varphi \right|\lt \frac{\pi }{2} \right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\left( \frac{\pi }{18},\frac{\pi }{3} \right)$. The graph of $f\left( x \right)$ is shifted to the left by $... | -1 | 0.375 | 7,617.4375 | 6,987 | 7,995.7 | |
Evaluate \[ \lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1 + x^{n+1}}{1 + x^n}\right)^{x^n}. \] | \frac{2}{e} | By taking logarithms, we see that the desired limit is $\exp(L)$, where $L = \lim_{x\to 1^-} \sum_{n=0}^{\infty} x^n \left( \ln(1+x^{n+1}) - \ln(1+x^n) \right)$. Now \begin{align*} &\sum_{n=0}^N x^n \left( \ln(1+x^{n+1}) - \ln(1+x^n) \right) \\ & = 1/x \sum_{n=0}^N x^{n+1} \ln(1+x^{n+1}) - \sum_{n=0}^N x^n\ln(1+x^n) \\... | 0 | 7,586.125 | -1 | 7,586.125 |
We wrote an even number in binary. By removing the trailing $0$ from this binary representation, we obtain the ternary representation of the same number. Determine the number! | 10 | 0 | 8,192 | -1 | 8,192 | |
Given $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha-\beta) = \frac{13}{14}$, with $0 < \beta < \alpha < \frac{\pi}{2}$, find $\beta$. | \frac{\pi}{3} | 0.875 | 5,456.8125 | 5,066.071429 | 8,192 | |
Given the sum of the first n terms of two arithmetic sequences, ${a_n}$ and ${b_n}$, are $S_n$ and $T_n$ respectively, and the ratio $\frac{S_n}{T_n} = \frac{2n+1}{3n+2}$, calculate the value of $\frac{a_3 + a_{11} + a_{19}}{b_7 + b_{15}}$. | \frac{129}{130} | 0.6875 | 6,167.1875 | 5,873.272727 | 6,813.8 | |
Emily ordered her playing cards by suit in the order $$A,2,3,4,5,6,7,8,9,10,J,Q,K,A,2,3,\cdots.$$What is the $42$nd card? | 3 | 0.9375 | 4,357.75 | 4,102.133333 | 8,192 | |
Let $Q(x) = x^2 - 4x - 16$. A real number $x$ is chosen at random from the interval $6 \le x \le 20$. The probability that $\lfloor\sqrt{Q(x)}\rfloor = \sqrt{Q(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + ... | 17 | 0 | 8,149 | -1 | 8,149 | |
There are 16 different cards, including 4 red, 4 yellow, 4 blue, and 4 green cards. If 3 cards are drawn at random, the requirement is that these 3 cards cannot all be of the same color, and at most 1 red card is allowed. The number of different ways to draw the cards is \_\_\_\_\_\_ . (Answer with a number) | 472 | 0 | 7,450.25 | -1 | 7,450.25 | |
How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ? | 120 | 0.125 | 7,818 | 5,206.5 | 8,191.071429 | |
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression, and the common difference $d$ is a multiple of $5$? | 11 | 0.3125 | 6,528.875 | 5,499.4 | 6,996.818182 | |
The sum of four positive integers that form an arithmetic sequence is 58. Of all such possible sequences, what is the greatest possible third term? | 19 | 1 | 3,175.1875 | 3,175.1875 | -1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.